Thom–Sebastiani Theorem

In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ ${\displaystyle f:(\mathbb {C} ^{n_{1}+n_{2}},0)\to (\mathbb {C} ,0)}$ defined as ${\displaystyle f(z_{1},z_{2})=f_{1}(z_{1})+f_{2}(z_{2})}$ where ${\displaystyle f_{i}}$ are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of ${\displaystyle f}$ is isomorphic to the tensor product of those of ${\displaystyle f_{1},f_{2}}$.^[1] Moreover, the isomorphism respects the monodromy operators in the sense: ${\displaystyle T_{f_{1}}\otimes T_{f_{2}}=T_{f}}$.^[2]

The theorem was introduced by Thom and Sebastiani in 1971.^[3]

Observing that the analog fails in positive characteristic, Deligne suggested that, in positive characteristic, a tensor product should be replaced by a (certain) local convolution product.^[2]